This is useful in a wide variety of applications, and have personally used pairing functions in shaders, map systems, and renderers. Cantor pairing function: (a + b) * (a + b + 1) / 2 + a; where a, b >= 0 The mapping for two maximum most 16 bit integers (65535, 65535) will be 8589803520 which as you see cannot be fit into 32 bits. \end{array} An Elegant Pairing Function Matthew Szudzik Wolfram Research Pairing functions allow two-dimensional data to be compressed into one dimension, and they play important roles in the arrangement of data for exhaustive searches and other applications. In: Wolfram Research (ed.) Abstract This article surveys the known results (and not very well-known re- sults) associated with Cantor’s pairing function and the Rosenberg-Strong pairing function, including their inverses, their generalizations to higher dimensions, and a discussion of a few of the advantages of the Rosenberg- Strong pairing function over Cantor… Viewed 40 times 0. The function outputs a single non-negative integer that is uniquely associated with that unordered pair. On Windows this value is uint16_t. a^2 + a + b & : a \ge b They may also differ in their performance. The foundation of efficient enumeration lies in both direct access to encoded structures, achieved with well-known and new pairing functions, and dependent enumeration, which embeds constraints into the enumeration to avoid backtracking. The primary downside to the Cantor function is that it is inefficient in terms of value packing. The performance between Cantor and Szudzik is virtually identical, with Szudzik having a slight advantage. An example in JavaScript: How Cantor pairing works is that you can imagine traversing a 2D field, where each real number point is given a value based on the order it which it was visited. In a more pragmatic way, it may be … Sometimes you have to encode reversibly two (or more) values onto a single one. In theoretical computer science they are used to encode a function defined on a vector of natural numbers : → into a new function : → 39. The pairing function can be understood as an ordering of the points in the plane. The proc gen seed is the gym number listed at the top of the board, which feeds into a Szudzik pairing function. -c - 1 & : (a < 0 \cap b \ge 0) \cup (a \ge 0 \cap b < 0) For example, extracting the n th element from our enumeration of three-tuples indexes about $\sqrt[3]{n}$ elements into each of its components instead of, say, indexing $\sqrt[2]{n}$ into one and $\sqrt[4]{n}$ into the other two, as you would if a three-tuple were built out of nested pairs. \right.$$, $$c(a,b) = \left\{\begin{array}{ll} Szudzik M (2006) An elegant pairing function. Additional space can be saved, giving improved packing efficiency, by transferring half to the negative axis. As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. Parameters [in] _a: First value, must be a non-negative integer. /// /// So, if user didn't make something stupid like overriding the GetHashCode() method with a constant, /// we will get the same unique number for the same row and column every time. Value. Ask Question Asked 1 year, 2 months ago. That fiddle makes note of the following references: $$index = \left\{\begin{array}{ll} 5 0 obj Download ElegantPairing.nb. %�쏢 So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. However, cantor(9, 9) = 200. x^2 + x + y & : x \ge y Different pairing functions known from the literature differ in their scrambling behavior, which may impact the hashing functionality mentioned in the question. - pelian/pairing Matthew P. Szudzik. , the unpairing function is defined as follows (for Cantor's unpairing function please refer to (Wolfram and Gad-el-Hak, 2003) and (Cantor, 1878)): (14) p f S z u d z i k − 1 (a) = a − ⌊ a ⌋ 2 x ⌊ a ⌋ y if x < y, or (15) p f S z u d z i k − 1 (a) = ⌊ a ⌋ x a − ⌊ a ⌋ 2 − ⌊ a ⌋ y. else. $$b = \left\{\begin{array}{ll} function(x, y, z) { max = MAX(x, y, z) hash = max^3 + (2 * max * z) + z if (max == z) hash += MAX(x, y)^2 if (y >= x) hash += x + y else hash += y return hash} This pairing function only works with positive numbers, but if we want to be able to use negative coordinates, we can simply add this to the top of our function: x = if x >= 0 then 2 * x else -2 * x - 1 The pairing function then combines two integers in [0, 2 26 -2] into a single integer in [0, 2 52). \end{array} This graphics demonstrates the path that Szudzik takes over the field: The primary benefit of the Szudzik function is that it has more efficient value packing. I used Matthew Szudzik's pairing function and got this: $(p - \lfloor\sqrt{p}\rfloor^2)\cdot\lfloor\sqrt{p}\rfloor = n$ -2x - 1 & : x < 0\\ Szudzik's Elegant Pairing Function. However, a simple transformation can be applied so that negative input can be used. A pairing function for the non-negative integers is said to be binary perfect if the binary representation of the output is of length 2k or less whenever each input has length k or less. The algorithms have been modified to allow negative integers for tuple inputs (x, y). It is always possible to re-compute the pair of arguments from the output value. Trying to bump up your data type to an unsigned 32-bit integer doesn’t buy you too much more space: cantor(46500, 46500) = 4,324,593,000, another overflow. /// 2- We use a pairing function to generate a unique number out of two hash codes. 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